Download Turbulent transport coefficients and residual energy in mean

PHYSICS OF PLASMAS 15, 022302 共2008兲
Turbulent transport coefficients and residual energy in mean-field
dynamo theory
Fujihiro Hamba1,a兲 and Hisanori Sato2
1
Institute of Industrial Science, University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8505, Japan
Japan Patent Office, Kasumigaseki, Chiyoda-ku, Tokyo 100-8915, Japan
2
共Received 3 December 2007; accepted 14 January 2008; published online 25 February 2008兲
The turbulent electromotive force in the mean-field equation needs to be modeled to predict a
large-scale magnetic field in magnetohydrodynamic turbulence at high Reynolds number. Using a
statistical theory for inhomogeneous turbulence, model expressions for transport coefficients
appearing in the turbulent electromotive force are derived including the ␣ coefficient and the
turbulent diffusivity. In particular, as one of the dynamo effects, the pumping effect is investigated
and a model expression for the pumping term is obtained. It is shown that the pumping velocity is
closely related to the gradient of the turbulent residual energy, or the difference between the
turbulent kinetic and magnetic energies. The production terms in the transport equation for the
turbulent electromotive force are also examined and the validity of the model expression is assessed
by comparing with earlier results concerning the isotropic ␣ coefficient. The mean magnetic field in
a rotating spherical shell is calculated using a turbulence model to demonstrate the pumping
effect. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2839767兴
I. INTRODUCTION
The generation of a large-scale magnetic field by a turbulent flow of an electrically conducting fluid is an important
problem in astrophysics and plasma physics. In general, turbulent velocity fluctuations enhance the effective magnetic
diffusivity. Dynamo action is necessary to generate and sustain a large-scale magnetic field against the turbulent diffusivity. The ␣ effect in the mean-field dynamo theory is well
known as one of the dynamo mechanisms and has been applied to solar and Earth’s magnetic fields.1–3 The ␣ dynamo
was also invoked to explain the sustainment of the magnetic
field in controlled fusion devices such as the reversed field
pinch 共RFP兲.4,5 From the physical point of view, it is very
interesting and important to investigate the dynamo effect
commonly seen in various magnetohydrodynamic 共MHD兲
turbulent flows.
Fundamental properties of MHD turbulence have been
studied theoretically and numerically. Owing to the effect of
Alfvén waves, the energy spectrum of isotropic homogenous
MHD turbulence is expected to show k−3/2 wavenumber dependence in contrast to the k−5/3 spectrum of non-MHD
turbulence.6,7 The spectra of kinetic and magnetic energies
have been investigated using turbulence theories such as the
eddy-damped quasinormal Markovian approximation and the
Lagrangian renormalized approximation.8–10 Moreover, direct numerical simulation 共DNS兲 of MHD turbulence is a
powerful tool to examine the energy spectrum.11–14 Haugen
et al.13 showed that the asymptotic spectrum is suggested to
be k−5/3. Mason et al.14 evaluated the velocity and magneticfield alignment to provide an explanation for the k−3/2 spectrum in the plane perpendicular to the guiding magnetic field.
Dynamo action such as the ␣ effect was also studied
using DNS.15–20 Brandenburg15 carried out a DNS of isotroa兲
Electronic mail: [email protected].
1070-664X/2008/15共2兲/022302/12/$23.00
pic helical turbulence using a helical forcing to estimate the
coefficients of the ␣ effect and of the turbulent diffusivity.
Ossendrijver et al.16 performed a DNS of magnetoconvection
to examine the dependence of the ␣ coefficient on the rotation and the magnetic field strength. In addition to the ␣
effect, it is known that the pumping effect term can be derived from the tensorial form ␣ijB j for the turbulent electromotive force where B j is the mean magnetic field.3 The mean
magnetic field can be transported through the turbulence medium with the pumping velocity even though there is no
mean motion. The ␣ effect and the pumping effect originate
in the isotropic and antisymmetric components of ␣ij, respectively. Rädler et al.21 and Brandenburg and Subramanian22
theoretically investigated turbulent transport coefficients including the pumping effect. The coefficients of the pumping
effect were also evaluated using DNS.17,20
In order to predict the mean magnetic field in realistic
MHD turbulent flows, it is necessary to evaluate the turbulent transport coefficients such as the ␣ coefficient and the
turbulent diffusivity. For a specific problem such as the solar
magnetic field, some appropriate spatial distribution of the
transport coefficients can be prescribed. However, in developing a general MHD turbulence model, model expressions
for the transport coefficients are necessary to close the system of mean-field equations. For non-MHD turbulence, the
Reynolds stress is often modeled using the eddy viscosity
approximation to predict the mean velocity field.23 To evaluate the turbulent viscosity, two-equation models have been
widely used such as the K-␧ model that treats the transport
equations for the turbulent kinetic energy K and its dissipation rate ␧. A statistical theory called the two-scale directinteraction approximation 共TSDIA兲 was developed to theoretically derive and improve the eddy viscosity model for
non-MHD turbulence.24 The TSDIA was shown to successfully derive a nonlinear eddy-viscosity model. For MHD tur-
15, 022302-1
© 2008 American Institute of Physics
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Phys. Plasmas 15, 022302 共2008兲
F. Hamba and H. Sato
bulence, a few attempts including the TSDIA have been
made to construct a model for transport equations.25–29 The
TSDIA is expected to be useful for deriving a universal
MHD turbulence model.
Using the TSDIA, Yoshizawa26 examined the pumping
effect and the turbulent diffusivity; he extended the K-␧
model to MHD turbulence where K is the sum of the turbulent kinetic and magnetic energies. In his work, the ␣ effect
was not examined because the ␣ coefficient is a pseudoscalar
and cannot be expressed in terms of scalars K and ␧ only.
Yoshizawa and Hamba30 paid attention to the turbulent residual helicity as a pseudoscalar in modeling the ␣ coefficient; they proposed a three-equation model by adopting the
turbulent residual helicity as a third model variable. The
three-equation model was used to simulate the magnetic
fields in the RFP 共Ref. 31兲 and in the Earth.32 In addition to
the ␣ effect parallel to the mean magnetic field, Yoshizawa33
derived a new dynamo term called the cross-helicity dynamo
parallel to the mean vorticity. The cross-helicity dynamo has
been applied to several turbulent flows in astrophysical and
engineering phenomena.23,34–38
Recently, Yokoi39 paid attention to the turbulent residual
energy, the difference between the turbulent kinetic and magnetic energies, to investigate the solar-wind turbulence. A
turbulence model for the turbulent residual energy equation
was solved to evaluate the radial dependence of the turbulent
quantities in solar wind.40 However, the effect of the turbulent residual energy on the turbulent electromotive force has
not been fully explored yet. It must be interesting to investigate the mean-field dynamo theory in more detail using the
TSDIA and to examine the effect of the turbulent residual
energy.
In this work, using the TSDIA we theoretically calculate
a model expression for the turbulent electromotive force. In
the present analysis, the formulation is improved so that the
frame-invariance of the model expression under rotating
transformations can be satisfied.41 The Green’s functions that
represent the response of the velocity and magnetic-field
fluctuations to a disturbance at a previous time are treated in
more accurate manner. In the perturbation expansion of the
TSDIA, we calculate a few additional terms for the turbulent
electromotive force compared to the previous work.23 We
derive the ␣ effect, the pumping effect, the turbulent diffusivity, and the cross-helicity dynamo effect, and we examine
the transport coefficients appearing in these terms.
This paper is organized as follows: In Sec. II, we explain
the turbulent dynamo and diffusivity terms appearing in
the turbulent electromotive force. In Sec. III, we apply the
TSDIA to derive a model expression for the turbulent electromotive force. In Sec. IV, we compare the present result
with the previous work. To assess the validity of the model
expression, we examine the transport equation for the turbulent electromotive force. To demonstrate the pumping effect,
we apply the model to the simulation of the magnetic field in
a rotating spherical shell. Concluding remarks will be given
in Sec. V.
II. MEAN-FIELD EQUATIONS AND DYNAMO EFFECTS
A. Mean-field equations
In this paper we adopt Alfvén velocity units and replace
the magnetic field b / 冑␳␮0 → b, the electric current density
j / 冑␳ / ␮0 → j, and the electric field e / 冑␳␮0 → e, where ␳ is
the fluid density and ␮0 is the magnetic permeability. The
Navier-Stokes equation for a viscous, incompressible, electrically conducting fluid and the induction equation for the
magnetic field in a rotating system are written as follows:
⳵u
= − ⵜ · 共uu − bb兲 − ⵜpM + ␯ⵜ2u − 2⍀F ⫻ u,
⳵t
共1兲
ⵜ · u = 0,
共2兲
⳵b
= − ⵜ ⫻ e,
⳵t
e = − u ⫻ b + ␭M j,
共3兲
ⵜ · b = 0.
共4兲
Here, u is the velocity, pM 共=p + 共⍀F ⫻ x兲 / 2 + b / 2兲 is the
total pressure, ⍀F is the system rotation rate, ␯ is the kinematic viscosity, ␭ M is the magnetic diffusivity, and j = ⵜ ⫻ b.
We use ensemble averaging 具·典 to divide a quantity into the
mean and fluctuating parts as
2
f = F + f ⬘,
F ⬅ 具f典,
2
共5兲
where f = 共u , b , p M , e , j , ␻兲 and ␻共=ⵜ ⫻ u兲 is the vorticity.
The evolution equations for the mean velocity U and the
mean magnetic field B are written as
⳵U
= − ⵜ · 共UU − BB兲 − ⵜ · R − ⵜPM + ␯ⵜ2U
⳵t
− 2⍀F ⫻ U,
ⵜ · U = 0,
⳵B
= − ⵜ ⫻ E,
⳵t
共6兲
共7兲
E = − U ⫻ B − EM + ␭ M J,
ⵜ · B = 0,
共8兲
共9兲
where Rij共=具ui⬘u⬘j − bi⬘b⬘j 典兲 is the Reynolds stress and
E M 共=具u⬘ ⫻ b⬘典兲 is the turbulent electromotive force. In order
to calculate the time evolution of the mean velocity and the
mean magnetic field, we need to evaluate Rij and E M ; some
modeling is necessary for the two quantities. Many turbulence models for the Reynolds stress have been developed
for non-MHD turbulence and some of them can be extended
to MHD turbulence. On the other hand, the turbulent electromotive force is treated only in MHD turbulence. In this
paper we focus on modeling the turbulent electromotive
force to investigate the dynamo effect.
B. Turbulent electromotive force
In the mean-field dynamo theory, the turbulent electromotive force can be expressed in a tensorial form
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Phys. Plasmas 15, 022302 共2008兲
Turbulent transport coefficients…
E Mi = ␣ijB j + ␤ijk
⳵B j
,
⳵xk
共10兲
where summation convention is used for repeated indices. If
isotropic components of the coefficients
␣ij = ␣␦ij,
␤ijk = ␤␧ijk
共11兲
共where ␦ij is the Kronecker delta symbol and ␧ijk is the unit
alternating tensor兲 are considered, the turbulent electromotive force can be written as
E M = ␣B − ␤J.
共12兲
The first term on the right-hand side of Eq. 共12兲 represents
the ␣ effect, whereas ␤ in the second term is the turbulent
diffusivity. For homogeneous turbulence an expression for
the ␣ coefficient was derived using the spectrum of the kinetic helicity 具u⬘ · ␻⬘典 in the wavenumber space in the meanfield theory.2,3 Pouquet et al.8 showed that the ␣ coefficient
can be expressed in terms of the spectrum of the turbulent
residual helicity H共=具−u⬘ · ␻⬘ + b⬘ · j⬘典兲. The ␣ coefficient
was also evaluated in the DNS of isotropic helical
turbulence.15–17,20
In general, the coefficient ␣ij can be decomposed into
the symmetric and antisymmetric components
␣Sij = 共␣ij + ␣ ji兲/2,
␣Aij = 共␣ij − ␣ ji兲/2.
共13兲
The antisymmetric component can be expressed in terms of a
vector ␣A defined as
␣Ai = − ␧ijk␣Ajk/2.
共14兲
If the antisymmetric component of ␣ij is incorporated in addition to the isotropic components, Eq. 共10兲 can be written as
E M = ␣B + ␣A ⫻ B − ␤J.
共15兲
The term ␣A ⫻ B represents the pumping effect; that is, the
mean magnetic field can be transported through the turbulence medium with the velocity ␣A even though there is no
mean motion.3 The pumping velocity ␣A is expected to be
proportional to −ⵜ具u⬘2典 and the pumping effect can be considered as the turbulence-induced diamagnetism.3 Using the
tau approximation Rädler et al.21 and Brandenburg and
Subramanian22 showed that the pumping velocity is proportional to the gradient of the difference between the kinetic
and magnetic energies, −ⵜ具u⬘2 − b⬘2典.
Using the TSDIA, Yoshizawa26 proposed a two-equation
MHD turbulence model. This model is an extension of the
non-MHD K-␧ model to MHD turbulence. As basic variables
the turbulent MHD energy K共=具u⬘2 + b⬘2典 / 2兲 and its dissipation rate ␧共=2␯具s⬘ijs⬘ij典 + ␭ M 具j⬘2典兲 were adopted where
s⬘ij关=共⳵ui⬘ / ⳵x j + ⳵u⬘j / ⳵xi兲 / 2兴 is the strain-rate fluctuation. The
turbulent electromotive force was modeled as
E M = ␣A ⫻ B − ␤J,
where
共16兲
␣ A = C ␣1
K
K2
ⵜ K − C␣2 2 ⵜ ␧,
␧
␧
␤ = C␤
K2
␧
共17兲
and C␣1, C␣2, and C␤ are nondimensional model constants.
The pumping velocity ␣A depends on the gradients of the
turbulent MHD energy and its dissipation rate. The turbulent
diffusivity ␤ is modeled in a similar form to the turbulent
viscosity for non-MHD turbulence. Using the pumping term,
Yoshizawa26 discussed the mechanism of the sustainment of
the mean magnetic field in the RFP. However, this model
does not involve the well-known ␣ effect because a pseudoscalar ␣ cannot be expressed in terms of scalars K and ␧
only.
Adopting the turbulent residual helicity H as a third
model variable, Yoshizawa and Hamba30 proposed a threeequation model for the ␣ effect. The ␣ coefficient in Eq. 共12兲
can be modeled in terms of H in a straightforward manner
because H is also a pseudoscalar. Since then, the ␣ effect has
been treated instead of the pumping effect in MHD studies
using the TSDIA. In addition, Yoshizawa33 pointed out the
importance of the cross helicity W共=具u⬘ · b⬘典兲 and proposed a
new dynamo term proportional to the mean vorticity ⍀. As a
result, the turbulent electromotive force in a rotating frame
can be modeled as23
E M = ␣ B − ␤ J + ␥ ⍀ + 2 ␥ F⍀ F ,
共18兲
where
K
␣ = C␣ H,
␧
␤ = C␤
K2
,
␧
K
␥ = ␥F = C␥ W.
␧
共19兲
The third and fourth terms on the right-hand side of Eq. 共18兲
represent the cross-helicity dynamo. In the TSDIA analysis,
the expressions for the transport coefficients appearing in Eq.
共18兲 are first derived in the wavenumber space; the forms of
␥ and ␥F are slightly different from each other. After the
one-point closure approximation, the two coefficients are
modeled in the same form as shown in Eq. 共19兲 and the
cross-helicity dynamo term can be written as ␥共⍀ + 2⍀F兲.
Since the turbulent electromotive force is objective, or
frame-invariant under rotating transformations, its model expression should also be objective.41 If ␥ = ␥F, the model expression given by Eq. 共18兲 is objective because the mean
absolute vorticity ⍀ + 2⍀F is objective. However, if ␥ ⫽ ␥F,
then Eq. 共18兲 cannot be objective. Therefore, at the beginning of the formulation the TSDIA must have some inaccurate procedure that violates the frame-invariance. In this
work we improve the TSDIA formulation to derive model
expressions satisfying the frame-invariance.
On the other hand, in Eq. 共19兲 the turbulence intensity is
expressed in terms of K, the sum of the turbulent kinetic and
magnetic energies. Recently, Yokoi39 pointed out the importance of the turbulent residual energy KR共=具u⬘2 − b⬘2典 / 2兲 in
the solar-wind turbulence and investigate its evolution equation. A four-equation model treating K, ␧, W, and KR was
proposed and solved to evaluate turbulent statistics in the
solar wind.40 In the present work, we will examine the relation of the residual energy to the turbulent electromotive
force.
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Phys. Plasmas 15, 022302 共2008兲
F. Hamba and H. Sato
III. THEORETICAL FORMULATION
⳵ui⬘
D̄ui⬘ ⳵ui⬘
=
+ Uj
+ ␧ jik⍀0ku⬘j .
⳵X j
DT
⳵T
⍀ 0 = ⍀ F/ ␦ ,
共24兲
In this section we briefly explain the procedure of the
TSDIA for deriving a model expression for the turbulent
electromotive force. Compared with the previous TSDIA, we
improve the formulation in the two respects: the treatment of
the rotating frame and of the Green’s functions. In the previous method, the fluctuating field is expanded in terms of
the mean-field gradient such as ⳵Ui / ⳵x j and of the system
rotation rate ⍀F independently. This independent treatment
leads to the difference between the forms of ␥ and ␥F in the
wavenumber space as will be shown in Sec. IV. In the
present method, we expand the fluctuating field so that the
mean field can always take an objective form such as the
mean absolute vorticity. In addition, in the previous formulation, two Green’s functions Ĝ␾␾ and Ĝ␺␺ are treated where
␾共=u + b兲 and ␺共=u − b兲 are Elsässer variables. The Green’s
function Ĝ␾␾ 共Ĝ␺␺兲 represents the response of ␾ 共␺兲 to a
disturbance in the ␾ 共␺兲 equation. The Green’s functions
Ĝ␾␺ and Ĝ␺␾ are assumed to be zero although the cross
response can exist. In the present formulation, we adopt
primitive variables u and b and treat four Green’s functions
Ĝuu, Ĝbb, Ĝub, and Ĝbu. Although the choice of the variables
is arbitrary, the neglect of the Green’s functions for the cross
response may lead to a different result. We expect that the
present method can give more accurate expressions.
In addition, the material derivative term such as Dui⬘ / DT
= ⳵ui⬘ / ⳵T + U j⳵ui⬘ / ⳵X j violates the frame-invariance. To make
the term objective, we adopt the co-rotational derivative
D̄ui⬘ / DT defined as Eq. 共24兲. The co-rotational derivative of
a vector is objective and adequately describes an unsteady
behavior of the fluctuating field.41,42 As a result, each term on
the right-hand side of Eq. 共22兲 can be expressed in an objective form. The induction equation for bi⬘ can be derived in a
similar manner. This formulation guarantees the frameinvariance of model expressions derived later.
A. Two-scale variables and frame-invariance
B. Perturbation expansion and Green’s function
First, we introduce two space and time variables using a
scale-expansion parameter ␦ as
Using the Fourier transform with respect to ␰ we express
a fluctuating field f ⬘ as
X共= ␦x兲,
␰共=x兲,
␶共=t兲,
T共= ␦t兲.
共20兲
Here, the fast variables ␰ and ␶ describe the rapid variation
of the fluctuating field whereas the slow variables X and T
describe the slow variation of the mean field. A quantity f
can be written as
f = F共X;T兲 + f ⬘共␰,X; ␶,T兲.
共21兲
The equations for the velocity fluctuation ui⬘ can be written
as
⳵ui⬘
⳵2ui⬘
⳵ui⬘
⳵ p⬘
⳵
+ Uj
+
共u⬘j ui⬘ − b⬘j bi⬘兲 + M − ␯
⳵␰ j ⳵␰ j
⳵␰ j⳵␰ j
⳵␶
⳵␰i
− Bj
−
冋
冉
冊
冉
1 ⳵Ui ⳵U j
⳵Ui
+ ␧ jik⍀0k =
+
2 ⳵X j ⳵Xi
⳵X j
+
冉
冊
冊
1 ⳵Ui ⳵U j
−
+ 2␧ jik⍀0k .
2 ⳵X j ⳵Xi
共25兲
dkf共k,X; ␶,T兲exp关− ik · 共␰ − U␶兲兴.
共26兲
Hereafter, the dependence of f共k , X ; ␶ , T兲 on X and T is not
written explicitly. We expand the fluctuation f共k ; ␶兲 in powers of ␦. We also solve the fluctuation iteratively with respect
to the mean magnetic field B by assuming that B is small.23
The former and latter expansions are denoted by indices n
and m, respectively. For example, the velocity fluctuation can
be written as
⬁
⬁
␦ nunmi共k; ␶兲
兺
n=0 m=0
⳵bi⬘
⳵bi⬘
⳵Bi
⳵Ui
= ␦ b⬘j
− u⬘j
+ ␧ jik⍀0k + B j
⳵␰ j
⳵X j
⳵X j
⳵X j
册
D̄ui⬘
⳵ p⬘
⳵
−
共u⬘u⬘ − b⬘j bi⬘ − 具u⬘j ui⬘ − b⬘j bi⬘典兲 − M ,
DT ⳵X j j i
⳵Xi
⳵ui⬘
⳵ui⬘
+␦
= 0,
⳵Xi
⳵␰i
冕
f ⬘共␰,X; ␶,T兲 =
ui共k; ␶兲 = 兺
共22兲
where
We should note that the mean-field gradient such as ⳵Ui / ⳵X j
is of the order ␦. To keep the terms involving the mean
velocity gradient objective, we assume that the system rotation rate is also of the order ␦ and is set to ⍀F = ␦⍀0. The
parenthesis involving the mean velocity gradient in Eq. 共22兲
can then be written as the sum of the mean strain-rate and
absolute-vorticity tensors which are both objective,
共23兲
⬁
⬁
k
⳵
␦ n+1i 2i
unmj共k; ␶兲,
兺
k ⳵XIj
n=0 m=0
−兺
共27兲
where
⳵
⳵
= exp共− ik · U␶兲
exp共ik · U␶兲.
⳵XIi
⳵Xi
共28兲
The second term on the right-hand side of Eq. 共27兲 is added
so that the continuity equation given by Eq. 共23兲 can be
satisfied. In this work, we expand the fluctuating field up to
the following order:
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022302-5
Phys. Plasmas 15, 022302 共2008兲
Turbulent transport coefficients…
ui共k; ␶兲 = u00i共k; ␶兲 + u01i共k; ␶兲 + ␦u10i共k; ␶兲
− ␦i
C. Calculation of turbulent electromotive force
ki ⳵
ki ⳵
u00j共k; ␶兲 − ␦i 2
u01j共k; ␶兲. 共29兲
2
k ⳵XIj
k ⳵XIj
The first through third terms on the right-hand side were
treated in the previous TSDIA, whereas the fourth and fifth
terms are newly examined in the present work.
Applying the Fourier transform to the evolution equations for u⬘ and b⬘, expanding the fluctuations as Eq. 共27兲,
and equating quantities in each order of ␦, we obtain the
equations for unmi共k ; ␶兲 and bnmi共k ; ␶兲. For example, the
equations for u01i共k ; ␶兲 and b01i共k ; ␶兲 are given by
⳵
u01i共k; ␶兲 + ␯k2u01i共k; ␶兲
⳵␶
− 2iM ijk共k兲
冕冕
The turbulent electromotive force can be written as
E Mi ⬅ ␧ijk具u⬘j bk⬘典 = ␧ijk
冕
dk具u j共k; ␶兲bk共k⬘ ; ␶兲典/␦共k + k⬘兲.
共37兲
From Eq. 共29兲, the correlation in the integrand in Eq. 共37兲 is
expanded as
具u jbk典 = 具u00jb00k典 + 具u01jb00k典 + 具u00jb01k典 + ␦具u10jb00k典
+ ␦具u00jb10k典 + ¯ .
The correlations of the basic fields u00i and b00i are assumed
to be isotropic and are written as
具␽00i共k; ␶兲␹00j共k⬘ ; ␶⬘兲典/␦共k + k⬘兲
关u00j共p; ␶兲u01k共q; ␶兲
pq
− b00j共p; ␶兲b01k共q; ␶兲兴 = − ik jB jb00i共k; ␶兲,
冕冕
关u00j共p; ␶兲b01k共q; ␶兲
− b00j共p; ␶兲u01k共q; ␶兲兴 = − ik jB ju00i共k; ␶兲,
␽␹
具Ĝ␽␹
ij 共k; ␶, ␶⬘兲典 = ␦ijG 共k; ␶, ␶⬘兲.
共31兲
respectively, where
=
dpdq␦共k − p − q兲,
共32兲
pq
1
M ijk共k兲 = 关k jDik共k兲 + kkDij共k兲兴,
2
Dij共k兲 = ␦ij −
k ik j
,
k2
共33兲
Nijk共k兲 = k j␦ik − kk␦ij .
共34兲
The right-hand sides of Eqs. 共30兲 and 共31兲 can be considered
as external forces for u01i共k ; ␶兲 and b01i共k ; ␶兲, respectively.
By introducing four Green’s functions for the velocity and
the magnetic field, we can formally solve the equations for
u01i共k ; ␶兲 and b01i共k ; ␶兲 as
u01i共k; ␶兲 = − ik jB j
冕
␶
b01i共k; ␶兲 = − ik jB j
冕
共40兲
For example, the spectra Quu and Huu correspond to the turbulent kinetic energy and the turbulent kinetic helicity, respectively, as
⬘2典/2 =
具u00
冕
dkQuu共k; ␶, ␶兲,
共41兲
冕
共42兲
⬘ · ␻00
⬘ 典=
具u00
dkHuu共k; ␶, ␶兲.
Substituting the formal solutions for the higher-order
terms up to the order given by Eq. 共29兲 into Eq. 共38兲, we can
express the turbulent electromotive force in terms of the basic correlations, the mean Green’s functions, and the mean
field. The resulting expression is given by
E M = ␣B − ␤J + ␥共⍀ + 2⍀F兲 − ⵜ␨ ⫻ B,
共43兲
where
+ I兵Gub,Hbu其兲,
Ĝub
ik 共k; ␶, ␶1兲u00k共k; ␶1兲兴,
␶
共39兲
␣ = 共1/3兲共− I兵Gbb,Huu其 + I兵Guu,Hbb其 − I兵Gbu,Hub其
d␶1关Ĝuu
ik 共k; ␶, ␶1兲b00k共k; ␶1兲
−⬁
+
i kk
␧ijkH␽␹共k; ␶, ␶⬘兲,
2 k2
where Q␽␹ and H␽␹ are basic correlations and ␽ and ␹ represent either u or b. The mean Green’s function is also expressed as
pq
冕 冕 冕冕
= Dij共k兲Q␽␹共k; ␶, ␶⬘兲 +
共30兲
⳵
b01i共k; ␶兲 + ␭ M k2b01i共k; ␶兲
⳵␶
− iNijk共k兲
共38兲
共35兲
␤⬘ = 共1/3兲共I兵Gbb,Quu其 + I兵Guu,Qbb其 − I兵Gbu,Qub其
− I兵Gub,Qbu其兲,
d␶1关Ĝbu
ik 共k; ␶, ␶1兲b00k共k; ␶1兲
共44兲
共45兲
−⬁
+ Ĝbb
ik 共k; ␶, ␶1兲u00k共k; ␶1兲兴.
共36兲
The governing equations for the Green’s functions are given
by Eqs. 共A3兲–共A6兲 in the Appendix. In a similar manner,
formal solutions for u10i共k ; ␶兲 and b10i共k ; ␶兲 can also be obtained 共see the Appendix for detail兲.
␥ = 共1/3兲共I兵Gbb,Qub其 + I兵Guu,Qbu其 − I兵Gbu,Quu其
− I兵Gub,Qbb其兲,
共46兲
␨ = 共1/3兲共I兵Gbb,Quu其 − I兵Guu,Qbb其 + I兵Gbu,Qub其
− I兵Gub,Qbu其兲,
共47兲
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022302-6
Phys. Plasmas 15, 022302 共2008兲
F. Hamba and H. Sato
␤ = ␤⬘ + ␨ = 共2/3兲共I兵Gbb,Quu其 − I兵Gub,Qbu其兲.
共48兲
Here, we introduce the following abbreviation of wavenumber and time integration:
I兵A,B其 =
冕 冕
␶
dk
d␶1A共k; ␶, ␶1兲B共k; ␶, ␶1兲.
共49兲
−⬁
The third term on the right-hand side of Eq. 共43兲 represents
the cross-helicity dynamo. In contrast to Eq. 共18兲, it is proportional to the mean absolute vorticity; the model expression derived at this stage is exactly objective. In addition, the
fourth term represents the pumping effect and the pumping
velocity is given by
␣A = − ⵜ␨ .
共50兲
As a result of the present stage of the TSDIA, the turbulent electromotive force is expressed in terms of wavenumber and time integrals of the basic correlations and the mean
Green’s functions. Such expressions are too complicated to
use in practical simulations; some simplification is necessary.
For non-MHD turbulence, specific forms of the basic correlations and the Green’s functions are often assumed using the
Kolmogorov spectrum in the wavenumber space and an exponential decay in time. For MHD turbulence, spectra of the
turbulent MHD energy and the turbulent residual helicity
have not been established yet although several theoretical
and numerical studies have been done. The form of the
Green’s functions is also difficult to assume. In future work,
some forms suggested by theoretical and numerical studies
should be substituted into Eqs. 共44兲–共48兲 to make full use of
the present result. Instead, in this work, we introduce the
following simplification to obtain a one-point closure model.
First, we assume that the Green’s functions satisfy
Guu共k; ␶, ␶⬘兲 = Gbb共k; ␶, ␶⬘兲,
Gub共k; ␶, ␶⬘兲 = Gbu共k; ␶, ␶⬘兲 = 0.
共51兲
Moreover, we replace the time integral with the Green’s
function by the turbulent time scale ␶T as
冕
␶
d␶1Guu共k; ␶, ␶1兲f共k; ␶, ␶1兲 = ␶T f共k; ␶, ␶兲,
−⬁
K
␶T = C␶ ,
␧
共52兲
where C␶ is a nondimensional constant. This simplification
was also made in the previous TSDIA.23 The resulting forms
can be easily integrated in wavenumber and expressed in
terms of the turbulent statistics in the physical space. As a
result, the transport coefficients appearing in Eq. 共43兲 are
expressed in terms of the turbulent statistics as follows:
K
␣ = C␣ H,
␧
K
K
␤ = C␤ 共K + KR兲 = 2C␤ Ku ,
␧
␧
共53兲
K
␥ = C␥ W,
␧
K
␨ = C␨ KR ,
␧
共54兲
where Ku = 具u⬘2典 / 2 and
C␣ = C␤ = C␥ = C␨共=C␶/3兲.
共55兲
Each transport coefficient in Eqs. 共53兲 and 共54兲 is expressed
as the product of the turbulent time scale K / ␧ and a characteristic statistical quantity such as H and Ku. The value of the
model constants C␣, C␤, C␥, and C␨ is expected to be approximately 0.1 and needs to be optimized using numerical
simulations.23
IV. DISCUSSION
A. Comparison with previous TSDIA
We compare the present result with the previous one in
more detail.23 The turbulent electromotive force derived in
the previous TSDIA was given in Eq. 共18兲. The transport
coefficients appearing in Eq. 共18兲 were expressed in terms of
wavenumber and time integrals as follows:
␣ = 共1/3兲共I兵GS,− Huu + Hbb其 − I兵GA,Hub − Hbu其兲,
共56兲
␤ = 共1/3兲共I兵GS,Quu + Qbb其 − I兵GA,Qub + Qbu其兲,
共57兲
␥ = 共1/3兲共I兵GS,Qub + Qbu其 − I兵GA,Quu + Qbb其兲,
共58兲
␥F = 共2/3兲共I兵GS,Qbu其 − I兵GA,Quu其兲.
共59兲
Here the Green’s functions for Elsässer variables are used;
they are related to those used in the present analysis as follows:
GS ⬅ 共G␾␾ + G␺␺兲/2 = 共Guu + Gbb兲/2,
共60兲
GA ⬅ 共G␾␾ − G␺␺兲/2 = 共Gub + Gbu兲/2.
共61兲
As mentioned before, the expression for ␥ given by Eq. 共58兲
is different from that for ␥F given by Eq. 共59兲. This defect is
cured by the introduction of the frame-invariant form of the
mean field in the present analysis.
In addition to the difference between ␥ and ␥F, the expression for the turbulent diffusivity is altered as shown in
Eqs. 共19兲 and 共53兲. We can see that besides the turbulent time
scale K / ␧, the turbulent diffusivity ␤ in Eq. 共19兲 is proportional to the turbulent MHD energy K共=Ku + Kb兲, whereas in
Eq. 共53兲 it is proportional to the turbulent kinetic energy Ku.
Therefore, if the turbulent magnetic energy Kb共=具b⬘2典 / 2兲 is
much greater than Ku as is expected in the Earth’s outer
core,43 the model expression 共19兲 may overestimate the turbulent diffusivity. Rädler et al.21 and Brandenburg and
Subramanian22 also showed that the turbulent diffusivity is
proportional to Ku. This was a consequence of a cancellation
arising from the pressure gradient term. This cancellation
corresponds to the addition of the ␨ term in Eq. 共48兲 in our
analysis.
The major difference between Eqs. 共18兲 and 共43兲 lies in
the pumping effect −ⵜ␨ ⫻ B appearing in Eq. 共43兲 only; this
term was not treated in the recent TSDIA.23 In the mean-field
dynamo theory,3 the pumping velocity ␣A was originally expected to be proportional to the gradient of the turbulent
kinetic energy, ⵜKu. In the K-␧ model of Yoshizawa,26 ␣A
has a term proportional to the gradient of the turbulent MHD
energy, ⵜK. In the present work, it is shown that ␣A is
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022302-7
Phys. Plasmas 15, 022302 共2008兲
Turbulent transport coefficients…
closely related to the gradient of the turbulent residual energy, ⵜKR, because ␨ is proportional to KR. The same result
was also obtained theoretically by Rädler et al.21 and Brandenburg and Subramanian.22 This result is expected to be
more appropriate in the following reason. Both the ␣ effect
and the pumping effect originate in the tensorial form ␣ijB j.
The ␣ coefficient corresponding to the isotropic component
of ␣ij was shown to be expressed in terms of the turbulent
residual helicity H, the difference between the kinetic and
current helicities.8 Therefore, it is natural that the pumping
velocity ␣A corresponding to the antisymmetric component
of ␣ij is expressed in terms of the turbulent residual energy,
the difference between kinetic and magnetic energies.
We should note that not only the pumping effect but also
the modification of the turbulent diffusivity are directly related to the quantity ␨ given by Eq. 共47兲. In Eq. 共48兲 the
coefficient ␤ is expressed as the sum of ␤⬘ and ␨; the coefficient ␤⬘ given by Eq. 共45兲 corresponds to ␤ given by
Eq. 共57兲 in the previous result. If ␨ is neglected in the present
result, the same expression as the previous result is recovered. In fact, the quantity ␨ stems from the fourth and fifth
terms in Eq. 共29兲 that are newly treated in the present
TSDIA.
B. Equation for turbulent electromotive force
Although a new model expression for the turbulent electromotive force is derived using the TSDIA, it is not yet
justified by numerical simulation or observation. In this subsection, to assess the validity of the model expression, we
examine the transport equation for the turbulent electromotive force. For non-MHD turbulence it is known that the
production terms in the transport equation for a correlation
such as the Reynolds stress are closely related to the eddyviscosity-type model for the correlation. For example, the
correlation can be modeled as the product of the production
terms and the turbulent time scale. A similar method called
the minimal tau approximation was used for MHD
turbulence.44,45 This type of approximation was also introduced in the TSDIA as the Markovian method.23 Therefore,
the investigation of the production terms gives a clue to
modeling the turbulent electromotive force.
The transport equation for E M in a rotating frame can be
written as
冉
冊
⳵
D̄E M
⬅
+ U · ⵜ + ⍀F ⫻ E M
Dt
⳵t
共62兲
where three production terms are given by
冉冓
⳵u⬘j
⳵xm
冔 冓 冔冊
+ bk⬘
⬘ 典 + 具b⬘j bm⬘ 典兲
PE2i = − ␧ijk共具u⬘j um
⳵b⬘j
⳵xm
⳵Bk
,
⳵xm
冉
冊
⳵Uk
+ ␧mkn⍀Fn ,
⳵xm
共65兲
and the detailed expression for the remaining part RE is
omitted. Each term in Eq. 共62兲 is expressed in an objective
form. The three production terms PE1, PE2, and PE3 involve
the mean velocity or the mean magnetic field; they represent
the effect of the mean field on the turbulent electromotive
force.
To show the correspondence of the production terms to
the terms in the model expression 共43兲, we first assume an
isotropic homogeneous turbulent field and replace the correlations appearing in Eqs. 共63兲–共65兲 as follows:
冓 冔
⳵␹⬘
1
具␽i⬘␹⬘j 典 = ␦ij具␽⬘ · ␹⬘典, ␽i⬘ k
3
⳵x j
1
= ␧ijk具␽⬘ · ⵜ ⫻ ␹⬘典.
6
共66兲
Then, we can obtain the following expressions:
PE1 = 31 HB,
PE2 = − 32 KJ,
PE3 = 32 W共⍀ + 2⍀F兲, 共67兲
which correspond to the first, second, and third terms on the
right-hand side of Eq. 共43兲, respectively. Therefore, it is
shown that the ␣ effect, the turbulent diffusivity, and the
cross-helicity dynamo results from the corresponding production terms.
The pumping effect term has not been obtained yet because the coefficient ⵜ␨ requires an inhomogeneous turbulent field. Next, instead of the isotropic homogeneous assumption 共66兲, we assume a turbulent field that is locally
isotropic and weakly inhomogeneous. Using the solenoidal
condition for ui⬘ and bi⬘ the production term PE1 can be exactly rewritten as
冉冓
冉冓
PE1i = ␧ipq − uk⬘
⳵u⬘p
⳵xk
− ␧kpq − uk⬘
冔 冓 冔冊
冔 冓 冔冊
⳵u⬘p
⳵xi
+ bk⬘
⳵b⬘p
⳵xk
+ bk⬘
⳵b⬘p
⳵xi
Bq
Bq .
共68兲
Although the summation convention is applied to repeated
indices including p, we consider here only the part in which
p = k. Then, we can see that the second parenthesis in Eq.
共68兲 does not contribute because of ␧kpq = 0 and the correlations in the first parenthesis can be approximated as
冓 冔
u⬘p
⳵u⬘p
⳵x p
=
1 ⳵
1 ⳵
具u⬘2典 ⯝
具u⬘2典.
2 ⳵x p p
6 ⳵x p
共69兲
As a result, the production term PE1 can be expressed as
= PE1 + PE2 + PE3 + RE ,
PE1i = Bm␧ijk − uk⬘
⬘ 典 + 具u⬘j bm⬘ 典兲
PE3i = ␧ijk共具b⬘j um
,
共63兲
共64兲
PE1 = − 31 ⵜ KR ⫻ B,
共70兲
which corresponds to the pumping effect term. Therefore,
not only the three terms already derived in the previous TSDIA but also the pumping effect term can be justified in the
sense that they have corresponding production terms in the
transport equation. In particular, the production term PE1
given by Eq. 共63兲 involves the difference between the velocity correlation 具uk⬘⳵u⬘j / ⳵xm典 and the magnetic-field correlation
具bk⬘⳵b⬘j / ⳵xm典; this fact accounts for the dependence of the
pumping effect on the turbulent residual helicity.
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022302-8
Phys. Plasmas 15, 022302 共2008兲
F. Hamba and H. Sato
FIG. 1. Pumping effect in a spherical region of large residual energy. Solid
lines represent the original uniform magnetic field and dashed lines stand for
the magnetic field transported by the pumping effect.
C. Pumping effect in rotating spherical shell
The physical meaning of the pumping effect is illustrated
in Fig. 1. Here we consider a uniform magnetic field B0
plotted by solid lines and a spherical region of large residual
energy given by
␨ = ␨0 exp共− r2/r20兲,
共71兲
where ␨ is defined as Eq. 共54兲. The pumping velocity −ⵜ␨ is
directed outward in the radial direction. The electromotive
force −ⵜ␨ ⫻ B0 and the resulting current ⌬J have a toroidal
component. As a result, the magnetic field is transported in
the outward direction as plotted by dashed lines. This is also
called the turbulence-induced diamagnetism because the
magnetic field in the central region is decreased.3
To demonstrate the behavior of the pumping effect in a
more realistic problem, we investigate the magnetic field in a
rotating spherical shell similar to the Earth’s outer core.
Hamba32 numerically simulated the axisymmetric mean
magnetic field in a rotating spherical shell using a Reynoldsaveraged MHD turbulence model. In the model, the turbulent
electromotive force is given by
E M = ␣B − ␤J,
共72兲
where
K
␣ = C␣ f ␣ H,
␧
␤ = C␤ f ␤
K2
.
␧
共73兲
Here, model constants are set to C␣ = C␤ = 0.09 and correction
coefficients f ␣ 共0 艋 f ␣ 艋 1兲 and f ␤ 共0 艋 f ␤ 艋 1兲 are introduced
to satisfy the realizability condition for the turbulent electromotive force, 兩E M 兩 艋 K. To concentrate on the estimate of the
FIG. 2. Profiles of the mean magnetic field in a rotating spherical shell
without pumping effect; 共a兲 poloidal field B p and 共b兲 toroidal field B␾.
pumping effect, the turbulent diffusivity ␤ is modeled in the
same form as that of Hamba;32 the modified expression 共53兲
is not adopted here. In addition to the induction equation for
the mean magnetic field, the transport equations for K, ␧, and
H are solved to evaluate the transport coefficients ␣ and ␤.
In the simulation, physical quantities are nondimensionalized
by the typical velocity U0共=5 ⫻ 10−4 m s−1兲, the radius of the
outer boundary rout共=3.48⫻ 106 m兲, and the fluid density
␳共=1.09⫻ 104 kg m−3兲. The magnetic field and the time
scale are then normalized by 冑␳␮0U0 = 0.585 G and rout / U0
= 221 year, respectively. The ratio of radii of the inner
and outer boundaries is set to rin / rout = 0.35. The outer regions at 0 ⬍ r ⬍ rin and at r ⬎ rout are assumed to be insulators. The angular velocity of the system rotation is set to
⍀F = 5 ⫻ 105. The time evolution of the magnetic field is calculated until a steady state is reached 共see Hamba32 for details兲.
First, we briefly explain the result of the simulation of
Hamba32 in which the pumping effect is not treated. Figure 2
shows the profiles of the mean magnetic field. The poloidal
field B p = 共Br , B␪兲 plotted in Fig. 2共a兲 is approximately a dipole field whereas the toroidal field B␾ plotted in Fig. 2共b兲 is
negative 共positive兲 in the northern 共southern兲 hemisphere. It
was shown in Hamba32 that the dipole field is sustained owing to the ␣2 dynamo; that is, the toroidal and poloidal fields
induce each other via the ␣ effect. Figure 3 shows the profiles of the turbulent MHD energy K and the turbulent residual helicity H. The turbulent MHD energy is produced by
the buoyancy effect. Since the value of K has a maximum at
r ⬃ rout and at ␪ = ␲ / 2, the gradient ⳵K / ⳵␪ is positive 共negative兲 in the northern 共southern兲 hemisphere. The transport
equation for H involves the production term −⍀F · ⵜK whose
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022302-9
Phys. Plasmas 15, 022302 共2008兲
Turbulent transport coefficients…
FIG. 3. Profiles of turbulent statistics in a rotating spherical shell without
pumping effect; 共a兲 turbulent MHD energy K and 共b兲 turbulent residual
helicity H.
value is positive 共negative兲 in the northern 共southern兲 hemisphere. As a result, the turbulent residual helicity H of the
same sign as −⍀F · ⵜK is produced as shown in Fig. 3共b兲.
The large value of H enhances the ␣ effect.
Next, to examine the pumping effect we add the pumping term to Eq. 共72兲 as follows:
E M = ␣B − ␤J − ⵜ␨ ⫻ B,
共74兲
where
K
␨ = C␨ f ␨ KR .
␧
共75兲
Here we adopt C␨ = 0.09 and f ␨ = f ␤. To accurately predict the
profile of KR, we need to solve its transport equation. However, its model equation including model constants has not
been established yet.40 In order to examine a qualitative effect of the pumping term, we assume here that KR is proportional to K as follows:
KR = − ␦RK,
共76兲
where ␦R is a small nondimensional constant; ␦R = 0.1 is
adopted here. A minus sign is included in Eq. 共76兲 because
KR is expected to be negative in the Earth’s outer core.43 We
should note that Eq. 共76兲 is just an assumption made for a
qualitative estimate; the transport equation for KR needs to be
modeled in future work.
To evaluate the effect of the pumping term −ⵜ␨ ⫻ B on
the magnetic field, we calculate the difference between the
original and newly obtained magnetic fields
FIG. 4. Magnetic-field difference due to the pumping effect with ␦R = 0.1;
共a兲 poloidal field ⌬B p and 共b兲 toroidal field ⌬B␾.
⌬B = B1 − B0 ,
共77兲
where B0 is the original field shown in Fig. 2 and B1 is the
field obtained by solving the same model as that of Hamba32
except for the pumping term with ␦R = 0.1.
Figure 4 shows the difference ⌬B between the two magnetic fields. The profile of the poloidal field plotted in Fig.
4共a兲 is simple; it is approximately a dipole field and the
direction is opposite to the original field. It is shown that the
pumping effect reduces the original dipole field in this case.
On the other hand, the toroidal field plotted in Fig. 4共b兲 is
rather complicated. In the region near the equator, the difference ⌬B is positive 共negative兲 in the northern 共southern兲
hemisphere; these signs are opposite to the corresponding
original field. In this region the toroidal field also decreases
owing to the pumping effect.
These profiles of ⌬B can be roughly explained by considering the directions of the pumping velocity −ⵜ␨ and of
the original magnetic field B0 as illustrated in Fig. 5. In Fig.
5共a兲 we consider the toroidal component of the pumping
term, 共−ⵜ␨ ⫻ B兲␾ in the region near the equator. The pumping velocity −ⵜ␨ points in the same direction as ⵜK, whereas
the poloidal field B p is directed opposite to the system rotation ⍀F. The pumping term 共−ⵜ␨ ⫻ B兲␾ is then positive in
this region. As a result, the pumping term induces a positive
toroidal current ⌬J␾ corresponding to the dipole field ⌬B p
plotted in Fig. 4共a兲. Therefore, the magnetic field ⌬B p is
induced in the same manner as the example illustrated in Fig.
1. On the other hand, in Fig. 5共b兲 we consider the poloidal
component of the pumping effect, 共−ⵜ␨ ⫻ B兲 p. The pumping
velocity −ⵜ␨ is in the positive 共negative兲 ␪ direction and the
toroidal field B␾ has a negative 共positive兲 value in the north-
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022302-10
Phys. Plasmas 15, 022302 共2008兲
F. Hamba and H. Sato
ern 共southern兲 hemisphere. The pumping term 共−ⵜ␨ ⫻ B兲 p
then points in the negative r direction in both the northern
and southern hemispheres and induces ⌬J p in the same direction. Such a current in the negative r direction near the
equator can account for the negative gradient of the toroidal
field, ⳵⌬B␾ / ⳵␪, in the region near the equator shown in Fig.
4共b兲. Therefore, in this simulation of a rotating spherical
shell, the pumping term has the effect of reducing the original magnetic field induced by the ␣ effect. In future work,
we need to solve the transport equation for KR to evaluate the
pumping effect more accurately.
V. CONCLUSIONS
To predict the mean magnetic field in MHD turbulence
at high Reynolds number, it is necessary to model the turbulent electromotive force. In this work, a new model expression for the turbulent electromotive force was derived using
the TSDIA. We improved the formulation of the TSDIA so
that obtained expressions can satisfy the frame-invariance
under rotating transformations. We adopted four Green’s
functions to take into account the effect of the mean field on
the fluctuations more accurately. The resulting expression for
the turbulent electromotive force consists of terms representing the ␣ effect, the turbulent diffusivity, the cross-helicity
dynamo, and the pumping effect. The transport coefficients
in the turbulent electromotive force are expressed in terms of
turbulent statistics such as the turbulent MHD energy and the
turbulent residual helicity. It was shown that the turbulent
diffusivity is proportional to the turbulent kinetic energy.
Moreover, the pumping velocity was shown to be closely
related to the gradient of the turbulent residual energy. These
results confirm earlier findings of Rädler et al.21 and Brandenburg and Subramanian22 using the tau approximation.
To assess the validity of the model expression, we examined the transport equation for the turbulent electromotive
force. Using the isotropic homogeneous assumption, the
three production terms were shown to correspond to the
terms representing the ␣ effect, the turbulent diffusivity, and
the cross-helicity dynamo. Using the weakly inhomogeneous
assumption, the pumping effect term was also derived from
one of the production terms. To demonstrate the pumping
effect, we simulated the magnetic field in a rotating spherical
shell. It was shown that in this case the pumping term has the
effect of reducing the magnetic field induced by the ␣ dynamo. In future work, the transport equation for the turbulent
residual energy should be solved to evaluate the pumping
effect more accurately. It is also important to validate the
pumping effect using data obtained from three-dimensional
simulations of MHD turbulence.
ACKNOWLEDGMENTS
F.H. would like to thank Dr. N. Yokoi for valuable discussion on the turbulent residual energy.
This work was partially supported by the Grant-in-Aid
for Scientific Research of Japan Society for the Promotion of
Science 共19560159兲.
FIG. 5. Pumping effect in a rotating spherical shell; 共a兲 poloidal component
and 共b兲 toroidal component.
APPENDIX: GOVERNING EQUATIONS FOR GREEN’S
FUNCTIONS
The equations for the basic fields u00i and b00i are written
as
⳵
u00i共k; ␶兲 + ␯k2u00i共k; ␶兲
⳵␶
− iM ijk共k兲
冕冕
关u00j共p; ␶兲u00k共q; ␶兲
pq
− b00j共p; ␶兲b00k共q; ␶兲兴 = 0,
共A1兲
⳵
b00i共k; ␶兲 + ␭ M k2b00i共k; ␶兲
⳵␶
− iNijk共k兲
冕冕
u00j共p; ␶兲b00k共q; ␶兲 = 0.
共A2兲
pq
Since these equations are the same as those for isotropic
turbulence, we assume that the basic fields u00i and b00i are
isotropic and that the anisotropic and inhomogeneous effects
can be incorporated through higher-order terms such as u01i
and u10i.
The equations for u01i and b01i are given by Eqs. 共30兲
and 共31兲, respectively. The left-hand sides of Eqs. 共30兲 and
共31兲 are linear with respect to u01i and b01i. The right-hand
sides of Eqs. 共30兲 and 共31兲 can be formally considered as
Downloaded 22 Apr 2008 to 133.11.199.19. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
022302-11
Phys. Plasmas 15, 022302 共2008兲
Turbulent transport coefficients…
⳵ ub
Ĝ 共k; ␶, ␶⬘兲 + ␯k2Ĝub
ij 共k; ␶, ␶⬘兲
⳵␶ ij
external forces for u01i and b01i. We then introduce the
Green’s functions satisfying the following system of equations:
− 2iM ikm共k兲
冕冕
ub
关u00k共p; ␶兲Ĝmj
共q; ␶, ␶⬘兲
pq
⳵ uu
Ĝ 共k; ␶, ␶⬘兲 + ␯k2Ĝuu
ij 共k; ␶, ␶⬘兲
⳵␶ ij
− 2iM ikm共k兲
冕冕
bb
共q; ␶, ␶⬘兲兴 = 0,
− b00k共p; ␶兲Ĝmj
⳵ bb
Ĝ 共k; ␶, ␶⬘兲 + ␭ M k2Ĝbb
ij 共k; ␶, ␶⬘兲
⳵␶ ij
uu
关u00k共p; ␶兲Ĝmj
共q; ␶, ␶⬘兲
pq
bu
共q; ␶, ␶⬘兲兴 = ␦ij␦共␶ − ␶⬘兲,
− b00k共p; ␶兲Ĝmj
共A5兲
共A3兲
− iNikm共k兲
冕冕
bb
关u00k共p; ␶兲Ĝmj
共q; ␶, ␶⬘兲
pq
ub
共q; ␶, ␶⬘兲兴 = ␦ij␦共␶ − ␶⬘兲.
− b00k共p; ␶兲Ĝmj
⳵ bu
Ĝ 共k; ␶, ␶⬘兲 + ␭ M k2Ĝbu
ij 共k; ␶, ␶⬘兲
⳵␶ ij
− iNikm共k兲
冕冕
bu
The Green’s functions Ĝuu
ij and Ĝij represent the response of
the velocity and the magnetic field, respectively, to a disturbance at time ␶⬘ in the velocity equation. On the other hand,
bb
Ĝub
ij and Ĝij represent the response to a disturbance in the
magnetic field equation. Using these Green’s functions, we
can obtain formal solutions for u01i and b01i given by Eqs.
共35兲 and 共36兲, respectively. In a similar manner, the solutions
for u10i and b10i can be written as
bu
关u00k共p; ␶兲Ĝmj
共q; ␶, ␶⬘兲
pq
uu
共q; ␶, ␶⬘兲兴 = 0,
− b00k共p; ␶兲Ĝmj
共A4兲
and
u10i共k; ␶兲 =
冕
␶
冋
d␶1Ĝuu
ij 共k; ␶, ␶1兲 D jk共k兲
−⬁
− D jk共k兲
b10i共k; ␶兲 =
册
D̄
u00k共k; ␶1兲 +
DTI
冕
␶
−⬁
册
冕
d␶1Ĝbu
ij 共k; ␶, ␶1兲 D jk共k兲
−⬁
− D jk共k兲
+ Bk
册
D̄
u00k共k; ␶1兲 +
DTI
冋
d␶1Ĝub
ij 共k; ␶, ␶1兲 − D jk共k兲
⳵
D̄
u00j共k; ␶1兲 − D jk共k兲
b00k共k; ␶1兲 ,
⳵XIk
DTI
冋
冉
冉
冊
⳵Bk
⳵Uk
u00m共k; ␶1兲 + D jk共k兲
+ ␧mkn⍀0n b00m共k; ␶1兲
⳵Xm
⳵Xm
共A7兲
冉
冊
⳵
⳵Bk
⳵Uk
b00m共k; ␶1兲 − D jk共k兲
+ ␧mkn⍀0n u00m共k; ␶1兲 + Bk
b00j共k; ␶1兲
⳵XIk
⳵Xm
⳵Xm
冕
␶
−⬁
冋
d␶1Ĝbb
ij 共k; ␶, ␶1兲 − D jk共k兲
册
⳵
D̄
u00j共k; ␶1兲 − D jk共k兲
b00k共k; ␶1兲 ,
⳵XIk
DTI
冉
冊
⳵Bk
⳵Uk
u00m共k; ␶1兲 + D jk共k兲
+ ␧mkn⍀0n b00m共k; ␶1兲
⳵Xm
⳵Xm
共A8兲
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共A9兲
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